6. Cathode Ray Tube (30 pts): Older television monitors used cathode ray tubes (CRT) to generate images on a screen. The CRT essentially steers an electron beam onto a phosphor coating that is located on the glass screen. By rapidly “painting" the beam across the phosphor screen, images (and hence movies) could be formed. Typically, the electron beam is deflected using magnetic fields, however, some designs also use electrostatic fields. In this problem you will analyze the electrostatic system.
The electron beam is deflected in the region 0 <= y <= d by two parallel plates with a potential difference Vo. For d <y<=d+L, there is no electric field present and the electron beam continues in a straight line based on the velocity it has at y = d.
a) Find an expression for the electric field E(y,z) between the two plates. *Hint: You can assume that the plates are infinitely large for calculation purposes.
b) Assume that at time t = 0 the electron beam starts at z(0) = y(0) = 0 with initial velocity v, (0) = v, and v,(0) = 0. Find an expression for the electron's coordinates z(t) and y(t). Assume the electron has charge q and mass m.
c) What is the vertical position, z ,of the electron when it exits the electric field region at y = d?
d) What is the velocity (v,(t) and v,(t)) of the electron when it exits the electric field region at y = d?
e) Finally, the electron beam is required to hit the phosphor screen at y = L +d, at a vertical position z = h. Find an expression for the voltage V(h) that is required such that the electron beam hits the phosphor screen at the desired height h.
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